Recent advances in sequencing technology has fundamentally transformed population genetics. Using whole-genome DNA sequence data, population geneticists now hope to estimate jointly many parameters of interest in complex models of evolution involving multiple populations. However, many of the statistical methods available for population genetic analyses are not scalable to the whole genome level. The main objective of the research proposed here is to develop a suite of mathematical, statistical, and computational methods that will allow researchers to more fully take advantage of the availability of genomic data. Likelihood-based approaches that take linkage information into account utilize more information in the data than do methods based on summary statistics and should therefore be more statistically efficient. However, they tend to require intensive computation, thus limiting their applicability. Aim 1 will develop a new statistical approach to ful-likelihood inference that can be applied at the genomic scale using many more sequences than previously possible. The distribution of segments of shared genetic similarity, i.e., segments of identity-by-descent or identity-by-state, contain important information about past demography and selection. Aims 2 and 3,will derive new theoretical results concerning such information and apply them to develop new statistical methods to tackle challenging problems such as the estimation of admixture proportions and admixture times, and inference of admixed DNA tracts. Recently, there has been much interest in using allele frequency spectra to estimate parameters in complex demography models. Aim 4 will develop efficient methods based on coalescent theory to compute the expected joint allele frequency spectra for more populations than could be previously considered. The use of the Wright-Fisher diffusion is ubiquitous in population genetics as a model for the forwards-in-time dynamics of the frequency of an allele in a large population. There are several population genetic applications in which it is natural to study the associated diffusion bridge. Aim 5 will investigate methods for simulating diffusion bridges in the presence of selection and obtain analytic results on the distribution of important functionals of the bridge path.